mathematics

Nash-Williams theorem

Nash-Williams theorem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Nash-Williams theorem rather than just read about it. In short: In graph theory, the Nash-Williams theorem is a tree-packing theorem that describes how many edge-disjoint spanning trees (and more generally forests) a graph can have: A graph G has t edge-disjoint spanning trees iff for every partition V 1 , … , V k ⊂ V ( G ) {\textstyle V_{1},\ldots ,V_{k}\subset V(G)} where V i ≠ ∅ {\displaystyle V_{i}\neq \emptyset } there are at least t(k − 1) crossing edges. The theorem was p…

Key takeaways

Nash-Williams theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Nash-Williams theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Nash-Williams theorem from memory before moving on to harder problems.

Reference excerpt

In graph theory, the Nash-Williams theorem is a tree-packing theorem that describes how many edge-disjoint spanning trees (and more generally forests) a graph can have:

A graph G has t edge-disjoint spanning trees iff for every partition V 1 , … , V k ⊂ V ( G ) {\textstyle V_{1},\ldots ,V_{k}\subset V(G)} where V i ≠ ∅ {\displaystyle V_{i}\neq \emptyset } there are at least t(k − 1) crossing edges.

The theorem was proved independently by Tutte and Nash-Williams, both in 1961. In 2012, Kaiser gave a short elementary proof. For this article, we say that such a graph has arboricity t or is t-arboric. (The actual definition of arboricity is slightly different and applies to forests rather than trees.)

Related tree-packing properties A k-arboric graph is necessarily k-edge connected. The converse is not true. As a corollary of the Nash-Williams theorem, every 2k-edge connected graph is k-arboric. Both Nash-Williams' theorem and Menger's theorem characterize when a graph has k edge-disjoint paths between two vertices.

Nash-Williams theorem for forests In 1964, Nash-Williams generalized the above result to forests:A graph G {\displaystyle G} can be partitioned into t {\displaystyle t} edge-disjoint forests iff for every U ⊂ V ( G ) {\displaystyle U\subset V(G)} , the induced subgraph G [ U ] {\displaystyle G[U]} has at most t ( | U | − 1 ) {\displaystyle t(|U|-1)} edges. Other proofs are given here. This is how people usually define what it means for a graph to be t-arboric.

In other words, for every subgraph S = G [ U ] {\displaystyle S=G[U]} , we have t ≥ ⌈ E ( S ) / ( V ( S ) − 1 ) ⌉ {\displaystyle t\geq \lceil E(S)/(V(S)-1)\rceil } . It is tight in that there is a subgraph S {\displaystyle S} that saturates the inequality (or else we can choose a smaller t {\displaystyle t} ). This leads to the following formula t = ⌈ max S ⊂ G E ( S ) V ( S ) − 1 ⌉ {\displaystyle t=\lceil \max _{S\subset G}{\frac {E(S)}{V(S)-1}}\rceil } ,also referred to as the Nash-Williams formula. The general problem is to ask when a graph can be covered by edge-disjoint subgraphs.

See also Arboricity Bridge (cut edge) Matroid partitioning Menger's theorem Tree packing conjecture

References

External links Paulson, Lawrence C. The Nash-Williams partition theorem (Formal proof development in Isabelle/HOL, Archive of Formal Proofs)

Worked examples

Example 1 — a first encounter with Nash-Williams theorem

Start with the simplest possible case. Write down what Nash-Williams theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Nash-Williams theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Nash-Williams theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Nash-Williams theorem

In research: Nash-Williams theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Nash-Williams theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Nash-Williams theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Theorems in graph theory, so understanding it makes those chapters shorter.
In everyday life: Look for Nash-Williams theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Nash-Williams theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Nash-Williams theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Nash-Williams theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Nash-Williams theorem in simple terms?

In graph theory, the Nash-Williams theorem is a tree-packing theorem that describes how many edge-disjoint spanning trees (and more generally forests) a graph can have: A graph G has t edge-disjoint spanning trees iff for every partition V 1 , … , V k ⊂ V ( G ) {\textstyle V_{1},\ldots ,V_{k}\subse…

Why does Nash-Williams theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Nash-Williams theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Nash-Williams theorem.